Exceptional characters and prime numbers in short intervals

Abstract

Under the assumption of the Riemann hypothesis the asymptotic value y/log x is known to hold for the number of primes in the short interval [x - y, x] for \(y = x^\alpha \) for every fixed \(\alpha < {1\over 2}\). We show under the assumption of the existence of exceptional Dirichlet characters the same asymptotic formula holds in the shorter intervals, for some \(\alpha < {1\over 2}\) \, in wide ranges of x depending on the characters.